1. Field of the invention
This invention relates to an improved high resolution Radon transform for use in analysing geophysical data.
2. Description of the Prior Art
The widespread use of multiple removal in the parabolic Radon domain (Hampson, 1986, Kostov, 1990) is related to its effectiveness and efficiency for most situations. However, when applied under severe conditions reduced spatial aperture (FIGS. 1a-f) and/or coarse offset sampling (FIGS. 2a-c), one may observe a poor focusing of the events in the parabolic Radon domain, combined with severe allasing artifacts. As a consequence, the multiple model loses some multiple energy and include significant primary energy. Once the multiple model is substracted from the input data, this leads to poor multiple removal and deteriorate primaries (see FIGS. 3a-d for illustration).
De-aliased Hiqh-Resolution (DHR) Radon Transform
Finite spatial aperture limits the resolution of the Radon transform, while finite spatial sampling introduces aliasing artifacts. To overcome these limitations one has to constrain the parabolic decomposition of the data. This issue was first investigated by Thorson and Claerbout, 1985. More recently Sacchi and Ulrych, 1995, Hugonner and Canadas, 1997, and Cary, 1998 have developed high-resolution Radon transforms (in the frequency-space or time-space domain). These constrain the Radon spectra to be sparse in q and t, using a re-welghted iterative approach.
However, such an iterative approach presents the major drawback of being very much time and computer consuming.
The present invention is directed toward a novel method that has the advantage of not being an iterative process.
Going back to the previous synthetic example (FIGS. 3a-c), the following observations can be made:
1. Due to the curvature range involved in this example and the parabolic sampling rate chosen, 550 parabolas are used to perform the parabolic decomposition of 15 traces. That is an under-determined least-squares problem.
2. Among the 550 parabolas, only 4 are actually needed to properly decompose the data. Constraining the parabolic decomposition of the data onto these four parabolas will lead to well-focused parabolic Radon spectra. One then has to solve a constrained underdetermined least-squares problem.
3. At low frequencies, the steering vectors used for the parabolic decomposition do not suffer from aliasing (FIG. 2b). As a consequence the parabolic decomposition at low frequencies can be used to guide the parabolic decomposition at higher frequencies.
To handle this constrained under-determined least-squares problem, the invention proposes a data driven constrained Radon decomposition. The Radon decomposition at a given frequency xcfx89k is constrained around the most significant spectral components highlighted at the previous frequency xcfx89kxe2x88x921. This non-iterative, gradual way (from low frequencies to high frequencies) to build the constrain enables to enhance the resolution of the Radon spectra. This algorithm enables to go beyond the commonly admitted sampling and aperture limitations. The task of the proposed method is simplified when the data to decompose are solely composed of a small amount of Radon components. On actual data this approach is therefor more effective using sliding temporal and spatial windows.
More generally, the invention proposes a method of performing a processing on seismic traces comprising the step of performing a constrained High Resolution Radon decomposition at various frequencies, wherein the Radon decomposition at a given frequency is constrained as a function of the Radon decomposition at at least a lower frequency.
More particularly, it proposes a method wherein a Radon decomposition is successively performed at various sparse frequencies, from the lower frequency to the higher, the Radon decomposition at a given frequency being constrained as a function of the Radon decomposition at the previous frequency.
Other features and advantages of the invention will be further understood in view of the following description.